arXiv:math-ph/0609009v1 5 Sep 2006 t ocntuttetno ed ntebs fteanalytica the of base the on fields tensor the construct m to Berwald-Moor ity the with space Finsler metric 4-dimensional h ikwk pc.Tebeko h nltct fteWorl p the limit. and, of non-relativistic space-time analyticity the 4-dimensional in the potential the of of break curving appear The non-trivial space Berwald-Moor space. The e Minkowski indicatrix function. the sugg World tangent we spac the the Here Riemannian and of space pseudo tensor broken. 4-dimensional 4-contravariant is the a analyticity base of this tensor when metric case the in also and oterlso eua counting. do world regular real of our for rules natural the most wo to the it is which and th system mathematics, numbers among algebraic of the are foundation complex numb multiplication the and principal presents the which the of real arithmetic associativity all rational, the of and integer, property commutativity natural, algebras, integral of contain the variety is that the au multiplication all the of the Although bases of conceptual algebras. the some these (and realize in commutativity paper the multiplication of the lack the of be num associativity) to complex seems 2-component theor this simple system relatively for the number reason of these theory t of the On to none even Nevertheless, number’. biquaternion forth. [1], is po searc so quaternions existing the a for and the to such obtained ’all but were of saying results plane success interesting Pythagoras the the famous on of the points case trust the In systems to number space-time. not E analogous the 4-dimensional correlated the on be for fractals could algebraic look which the researches of many harmony makes the It in example, for itself, osrcino h suoReana geometry Riemannian pseudo the of Construction h pc fteascaiecmuaiehprcmlxnumb complex hyper commutative associative the of space The h acntn euyo h hoyo h ucin fcmlxv complex of functions the of theory the of beauty fascinating The Introduction 1 ntebs fteBradMo geometry Berwald-Moor the of base the on amnSaeTcnclUiest,Moscow University, Technical State Bauman usa lcrtcnclIsiue Moscow Institute, Electrotechnical Russian [email protected] ..Garas’ko G.I. ..Pavlov D.G. [email protected] 1 uto fteBerwald-Moor the of quation riual,t h Newtonian the to articularly, ti.I rvdstepossibil- the provides It etric. ucin fthe of functions l saetm)uiga a as using (space-time) e ob lsl eae to related closely be to s ucinlast the to leads function d s a oconstruct to way a est e a ecompared be can ies h commutativity the ,w ol really could we h, 24,otvs[5] octaves [2-4], s sntcorrespond not es h lmnsof elements the , ie vno the of even times l esrneif strange be uld es h main The bers. ial,the Finally, . plane. uclidian ral reveals ariable ers, hr fthis of thors i a,the way, his xosof axioms e rsystems er H nso the of ints 4 H variable 4 sa is , One of the systems free from this drawback is the algebra of the commutative and associative hyper complex numbers, related to the direct sum of the four real algebras, which will be denoted as H4. The algebra of these numbers is isomorphic to the algebra of the 4-dimensional square real diagonal matrices, and the corre- sponding space is a linear Finsler space with the Berwald-Moor metric (the last fact was proved by the authors in [6]). It should be mentioned that Finsler space with the Berwald-Moor metric has been known and partially investigated for a long time [7–8]. One of the main properties of this space is the existence of such a range of the parameters that the 3-dimensional distances (from the point of view of the observer who uses the radar method to measure them [9]) correspond to the positively defined metric function the limit of which is the quadratic form [10]. In other words, the 3-dimensional world observed by an ”H4 inhabitant” is Euclidian within certain accuracy. Moreover, when one passes to the relativistic velocities, the 4-dimensional intervals between the H4 events present the Minkowski space correlations [11]. All this makes possible to suggest that the H4 space and the corresponding Finsler ge- ometry can be used as a mathematical model of the real space-time, and maybe this model would be even more productive than the pseudo Riemannian constructions prevailing in Physics now. Any hyper complex algebra is completely defined by the multiplication rule for the elements of a certain fixed basis. In the H4 number system there is a special – isotropic – basis e1, e2, e3, e4, such that

1 , if i = j = k , e e = pk e pk = (1) i j ij k ij   0 , else . Any analytical function in this basis can be given as

1 1 2 2 3 3 4 4 F (X)= f (ξ )e1 + f (ξ )e2 + f (ξ )e3 + f (ξ )e4 , (2) where H X = ξ1e + ξ2e + ξ3e + ξ4e , (3) 4 ∋ 1 2 3 4 and f i are four arbitrary smooth functions of a single real variable. In H4 there is one more – orthogonal – selected basis 1, j, k, jk, which is related to the isotropic basis by the following formulas

1= e1 + e2 + e3 + e4 ,  j = e1 + e2 e3 e4 , − −  (4)  k = e1 e2 + e3 e4 ,  − −  jk = e1 e2 e3 + e4 ,  − −   where 1 is the unity of algebra, and the corresponding component of the analytical function of the H4 variable is defined by the formula 1 u = f 1(ξ1)+ f 2(ξ2)+ f 3(ξ3)+ f 4(ξ4) . (5) 4  2  If X is a radius vector, then the coordinate space ξ1, ξ2, ξ3, ξ4 is a Berwald- Moor space with the length element

ds = 4 dξ1dξ2dξ3dξ4 4 g dξidξjdξkdξl , (6) ≡ ijkl p q where 1 , (i = j = k = l), 4! 6 6 6 gijkl =  (7)  0 , (else). For this geometry the tangent indicatrix equation is  gijklp p p p 1=0 , (8) i j k l − where 44 , (i = j = k = l), ijkl 4! 6 6 6 g =  (9)  0 , (else),

j k l  gijkldξ dξ dξ pi = (10) m r s t 3/4 (gmrstdξ dξ dξ dξ ) are the components of the generalized momentum or generalized momenta. k ijkl If we have tensors pij, gijkl, g and vector fields of the analytical func- tions F(A)(X) of the H4 variables, we could construct the metric tensors in the 4-dimensional space-time in many ways. For example,

k l gij(ξ)= gijklf(1)f(2) , (11) Now one can investigate the obtained Riemannian geometry. The main drawback of this approach is the variety of the ways to construct it. It is known [12] that if the tangent indicatrix equation is defined as

Φ(p; ξ)=0 , (12)

then the geodesics will be the solutions of the canonical system of differential equa- tions ∂Φ ∂Φ ξ˙i λ p ξ , p λ p ξ , = ( ; ) ˙i = i ( ; ) (13) ∂pi · −∂ξ · λ(p; ξ) = 0 is an arbitrary smooth function, and a dot above ξi and p means the 6 i derivation by the evolution parameter, τ.

2 Construction of the metric function of the pseudo Riemannian space

Let us regard a space which is conformally connected to the H4 space, that is to the space with the length element

ds′ = κ(ξ) 4 g dξidξjdξkdξl , (14) · ijkl q 3 where κ(ξ) > 0 is a scalar function which is a contraction-extension coefficient depending on the point. Let there be a normal congruence of geodesics (world lines). Then there is a scalar function S(ξ) (see, e.g. [12]) such that its level hyper surfaces are transversal to this normal congruence of the world lines and this function is a solution of the equation ∂S ∂S ∂S ∂S gijkl = κ(ξ)4 , (15) ∂ξi ∂ξj ∂ξk ∂ξl while the generalized momenta along this congruence of the world lines are related to S(ξ) by ∂S p = , (16) i ∂ξi

The equations for the world lines obtain the form

∂S ∂S ∂S ξ˙i = gijkl λ(ξ) , (17) ∂ξj ∂ξk ∂ξl ·

were λ(ξ) = 0. 6 In Physics the function S(ξ) is called ”action as a function of coordinates” and (15) is known as the Hamilton-Jacoby equation. In [10] the function S(ξ) was called the World function. If there is a congruence of the world lines, then the evolution of every point in space is known, particularly, the velocity field is known, but the energy char- acteristics of the material objects (observers) corresponding to a given world line are not known. The knowledge of the World function S(ξ) makes it possible to calculate the generalized momenta pi, corresponding to the energy characteristics, and the invariant energy characteristic, κ(ξ), which has also the meaning of the local contraction-extension coefficient of the plane H4 space. So, if our world view is the classical mechanics, then any pair out of the three: World function, congruence of the world lines, Finsler geometry - gives us the com- plete knowledge of the World. Let us construct a twice contravariant tensor gij(?) in the following way:

1 ∂S ∂S gij(ξ)= gijkl . (18) κ(ξ)4 · ∂ξk ∂ξl

Since 44 det(gij(ξ)) = =0 , (19) −33κ(ξ)8 6

then everywhere where the geometry (14) is defined, one can construct a tensor gij(ξ) such that ik i g (ξ)gkj(ξ)= δj , (20) 4 2 ∂S ∂S ∂S ∂S ∂S ∂S ∂S 2 1 1 2 1 3 1 4 − ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ    2  ∂S ∂S ∂S ∂S ∂S ∂S ∂S ∂ξ1 ∂ξ2 2 ∂ξ2 ∂ξ2 ∂ξ3 ∂ξ2 ∂ξ4 g (ξ)=4  −  . (21) ij    2  ·  ∂S ∂S ∂S ∂S ∂S ∂S ∂S   1 3 2 3 2 3 3 4   ∂ξ ∂ξ ∂ξ ∂ξ − ∂ξ ∂ξ ∂ξ     2   ∂S ∂S ∂S ∂S ∂S ∂S ∂S   1 4 2 4 3 4 2 4   ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ ∂ξ − ∂ξ   1 2 3 4   No doubt that in the same coordinate space ξ ,ξ ,ξ ,ξ such tensor gij(ξ) defines a Riemannian or pseudo Riemannian geometry with the length element

′′ i j ds = gij(ξ)dξ dξ . (22) q The construction of tensor gij(ξ) leads directly to the conclusion: the change of geometry (14) to the geometry (22) does not lead to the change of the initial congruence of the world lines and corresponding World function S(ξ). Therefore, in our concept one and the same World, i.e. the pair World func- tion; congruence of the world lines , corresponds to a whole class of{ related but } qualitatively different Finsler geometries.

3 Analyticity condition and the Minkowski space Let the World function S(ξ) be the (unity) component of an analytical function of the H4 variable in the orthogonal basis (4), that is 1 S(ξ)= f 1(ξ1)+ f 2(ξ2)+ f 3(ξ3)+ f 4(ξ4) . (23) 4   Then ∂S ∂S ∂S ∂S ∂f 1(ξ1) ∂f 2(ξ2) ∂f 3(ξ3) ∂f 4(ξ4) gijkl = = κ(ξ)4 > 0 , (24) ∂ξi ∂ξj ∂ξk ∂ξl ∂ξ1 ∂ξ2 ∂ξ3 ∂ξ4

and this leads to the limitation on the functions, fi:

∂f 1(ξ1) ∂f 2(ξ2) ∂f 3(ξ3) ∂f 4(ξ4) > 0 . (25) ∂ξ1 ∂ξ2 ∂ξ3 ∂ξ4

It follows from (24) that the space with the length element (14) can be obtained from the space with the length element (6) with the help of the conformal trans- formation, which means that the condition of the analyticity of the World function can be treated in a sense as the condition of the conformal symmetry. Let us construct tensor gij(ξ) following the algorithm developed in the previous section. It turns out that in a region where functions f i have no singularities there will always be such a coordinate system x0, x1, x2, x3 in which the length element ds′′ has a form ds′′ = (x0)2 (x1)2 (x3)2 (x3)2 . (26) − − − p 5 Let us express the coordinates x0, x1, x2, x3 in terms of the initial coordinates ξ1,ξ2,ξ3,ξ4:

1 x0 = f 1(ξ1)+ f 2(ξ2)+ f 3(ξ3)+ f 4(ξ4) , 4  √3
 x1 = f 1(ξ1)+ f 2(ξ2) f 3(ξ3) f 4(ξ4) ,   4 − −   (27) √3
 x2 = f 1(ξ1) f 2(ξ2)+ f 3(ξ3) f 4(ξ4) ,  4 − −
 √3  x3 = f 1(ξ1) f 2(ξ2) f 3(ξ3)+ f 4(ξ4) .  4 − −  
  Therefore, to obtain the non-trivial curving of the space-time one should use the World functions with the broken conformal symmetry.

4 Newtonian potential

Let us show that there are World functions that lead to the non-trivial pseudo Riemannian 4-dimensional spaces. Let us regard a function

1 S(ξ)= ξ1 + ξ2 + ξ3 + ξ4 + α ψ(̺) , (28) 4 ·
where α is the parameter characterizing the break of the analyticity of the World function (the break of the conformal symmetry in the H4 space), ψ is an arbitrary function of a single argument

̺ = (y1)2 +(y2)2 +(y3)2 , (29) p and y0, y1, y2, y3 are the coordinates in the orthogonal basis 1,j,k,jk:

1 y0 = (ξ1 + ξ2 + ξ3 + ξ4) , 4  1 1 1 2 3 4  y = (ξ + ξ ξ ξ ) ,  4 − −   (30) 1 1 2 3 4  y2 = (ξ ξ + ξ ξ ) ,  4 − −

1 1 2 3 4  y3 = (ξ ξ ξ + ξ ) .  4 − −    6  Then the derivatives of the World functions over the coordinates ξi can be expressed in the following way:

∂S 1 α dψ = 1+ y1 + y2 + y3 , ∂ξ1 4 ̺ d̺   
∂S 1 α dψ  = 1+ y1 y2 y3 ,  ∂ξ2 4 ̺ d̺ − −    
 (31) ∂S 1 α dψ  = 1+ y1 + y2 y3 ,  ∂ξ3 4 ̺ d̺ − −  
 ∂S 1 α dψ 1 2 3  = 1+ y y + y .  ∂ξ4 4 ̺ d̺ − −    
 Let us calculate the components of the metric tensor in coordinate s y0, y1, y2, y3 using the invariance of the square of the length element

i j i j gij(ξ)dξ dξ =g ˜ij(y)dy dy (32)

Grouping the terms, one gets

2 2 α 2 2 dψ 2 dψ 4(y ) g˜ =1 3α , g˜ − = 3 1+ α 1 , (33) 00 − d̺ ββ − d̺ − 3ρ2   (    )

dψ yβ dψ 2 y1y2y3 2g ˜ = 4 α +3α2 , (34) 0β − d̺ ̺ d̺ · yβ̺2 "   # dψ yδ dψ 2 yβyγ 2˜g = 4 3α + α2 , (35) βγ − d̺ ̺ d̺ · ̺2 "   # where β, γ, δ, = 1, 2, 3; β β− but no summation is performed here; in the last formula all the indices β, γ,≡ δ are different. If α = 0, then (˜g )= diag(1, 3, 3, 3) . (36) ij − − − This means that the real physical coordinates x0, x1, x2, x3 of the space-time are expressed by the coordinates y0, y1, y2, y3 in the following way

x0 = y0 , xβ = √3 yβ . (37) · Let us pass to the physical coordinates x0, x1, x2, x3:

i j i j g˜ij(y)dy dy =g ¯ij(x)dx dx , (38)

where 1 1 g¯00 =g ˜00 , g¯0β = g˜0β , g¯βγ = g˜βγ . (39) √3 · 3 · Let us denote r = (x1)2 +(x2)2 +(x3)2 √3 ̺ , (40) ≡ · p 7 Then

2 2 α 2 2 dψ 2 dψ 4(x ) g¯ =1 9α , g¯ − = 1+3α 1 , (41) 00 − dr ββ − dr − 3r2   (    )

dψ xβ dψ 2 x1x2x3 2g ¯ = 4 α +3√3α2 , (42) 0β − dr r dr · xβr2 "   # dψ xδ dψ 2 xβxγ 2¯g = 4 √3α + α2 . (43) βγ − dr r dr · r2 "   # 1 2 3 The metric tensorg ¯ij(x)=¯gij(x , x , x ) depends only on the space coordi- nates x1, x2, x3, and this corresponds to the stationary gravitational field, stationary Universe. The probe particle of mass m moves along the geodesic of the pseudo 1 2 3 Riemannian space with metric tensorg ¯ij(x , x , x ). Let a particle move in a fixed frame and have velocity much less than the light velocity, c: dxβ = vβ , vβ c , (44) dt | |≪ The gravitational fields are weak, that is the condition vβ << 1 remains valid | | for all the time of the particle motion. Let us obtain the Lagrange function, L, to describe such non-relativistic motion of the probe particle in the weak gravity field. To do this, develop the right hand side of the expression

g¯ (x1, x2, x3)dxidxj L = mc ij (45) dt − · p v 2 Within the accuracy of c
1 vβ vβvγ L = mc2√g¯ 1+ 2¯g +¯g , (46) − 00 · g¯ 0β c βγ c2 s 00  

1 vβ vβvγ 1 vβ 2 L mc2√g¯ 1+ 2¯g +¯g 2¯g . (47) ≃ − 00 · 2¯g 0β c βγ c2 − 8¯g2 0β c ( 00   00   ) Opening the brackets in the right hand side, we get an additive term which is the full time derivative of a certain function f(r), it depends linearly on the velocity components and, thus, it can be omitted. Leaving the same designation for the Lagrange function, we get

1 vβvγ 1 vβ 2 L mc2√g¯ 1+ g¯ 2¯g . (48) ≃ − 00 · 2¯g · βγ c2 − 8¯g2 0β c ( 00 00   ) Our goal is the Lagrange function of the form m~v2 L = U(~x) , (49) 2 − 8 where U(~x) is the potential energy of the probe particle, ~x (x1, x2, x3), ~v (v1, v2, v3), r2 = ~x 2, ~v 2 = (v1)2 +(v2)2 +(v3)2 v2. To reach≡ it we have to make≡ some assumptions about the correlation between≡ the parameter, α and light velocity: ν α = , when c α 0 . (50) c →∞ → v Besides, let α be of the same order (or smaller) with the relation . Then leaving c only the terms that don’t disappear at c in the (48), one gets →∞

9 ν2 dψ 2 v1v1 + v2v2 + v3v3 L mc2 + mc2 + m . (51) ≃ − 2 c2 dr · 2   Since ( mc2) is a full time derivative of function ( mc2 t), we omit it and get − − · m~v2 9mν2 dψ 2 L + . (52) ≃ 2 2 dr   Let a mass M be motionless in the frame origin, and then the potential energy of the probe particle with mass m located at x1, x2, x3 is equal to mM U(r)= γ , (53) − r where γ is the gravitational constant. Comparing (49) and (52), we get the equation for ψ(r):

9mν2 dψ 2 mM dψ √2γM 1 = γ = . (54) 2 dr r ⇒ dr ± 3ν r1/2   Therefore, 2√2γM ψ(r)= r1/2 + ψ (ψ = const). (55) ± 3ν · 0 0 Finally, the World function is equal to 2√2γM S = x0 r1/2 + C (C = const), (56) ± 3c · 0 0 When it performs a conformal transformation of the length element of the plane Berwald-Moor space, it induces a pseudo Riemannian geometry in the Minkowski space. For a non-relativistic probe particle of mass m, this geometry gives the motion equations for the Kepler problem for the point mass M located in the origin of the space frame. The more complicated World function, maybe also leading to the stationary Universe, has the form 1 S(ξ)= ξ1 + ξ2 + ξ3 + ξ4 [1 + α ψ (̺)] + α ψ (̺) , (57) 4 1 · 1 2 · 2
where αA are the parameters of the analyticity break of the World function (pa- rameters of the conformal symmetry break in the H4 space), ψA are the arbitrary functions of single argument ̺ (29), (30).

9 Conclusion The results obtained in this paper point at the deep correlation between the Einstein geometries and Finsler spaces with Berwald-Moor metric. We managed to find the concrete Finsler space with the Berwald-Moor metric which in the limit appeared to be related to the curved pseudo Riemannian space with the Newtonian gravitational potential. This fact points at the principal possibility to built more interesting constructions, particularly, such Finsler spaces whose limit cases would be the known relativistic solutions.

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